A general resistive, capacitive and inductive network can be analyzed by using the circuit laws known as Kirchoff's current law (KCL) and Kirchoff's voltage law (KVL) to generate sets of coupled differential equations. A common method to generate these equations is called Modified Nodal Analysis (MNA). Other methods such as the Sparse Tableau Method may also be used to create equivalent systems of equations (see, e.g., T. L. Pillage, R. A. Rohrer and C. Visweswariah, “Electronic Circuit and System Simulation Methods”, McGraw Hill, 1995).
Many methods, such as backward Euler, trapezoidal rule, etc. can be used to discretize the differential equations for transient simulation. The discretized differential equations represent the relation between the voltages and/or currents in the circuit at two or more time instances. Thus, from the known voltages and/or currents at one or more times instances, the voltages and/or currents at another time instance can be obtained. The differential equations are typically discretized in time to produce a matrix equation for a time step. Typically, the matrix equation for a linear circuit with inductors represents a linear equation system whose unknown is the vector of voltages and currents in the circuit at a time step. The known vector of the matrix equation (e.g., the right hand side of the equations of the linear equation system) typically depends on previous known time step results and/or known input stimuli.